Question

Statement I Two events $A$ and $B$ are called mutually exclusive events, if the occurrence of any one of them excludes the occurrence of the other event.
Statement II In the above statement, if they can not occur simultaneously, then the events $A$ and $B$ are not mutually exclusive.

• A. Statement I is true; Statement II is true; Statement II is a correct explanation for Statement I
• B. Statement I is true; Statement II is true; Statement II is not a correct explanation for Statement I
• C. Statement I is true; Statement II is false
• D. Statement I is false; Statement II is true

Answer 1

Answer: (B)

In the experiment of rolling a die, a sample space is $S=\{1,2,3,4,5,6\}$
Consider events $A$ 'an odd number appears' and $B$ 'an even number appears'.
Clearly, the event $A$ excludes the event $B$ and vice-versa. In other words, there is no outcome which ensures the occurrence of events $A$ and $B$ simultaneously. Here,
$A=\{1,3,5\}$ and $B=\{2,4,6\}$
Clearly, $A \cap B=\phi$, i.e., $A$ and $B$ are disjoint sets.
In general, two events $A$ and $B$ are called mutually exclusive events, if the occurrence of anyone of them excludes the occurrence of the other event, i.e., they cannot occur simultaneously. In this case, the sets $A$ and $B$ are disjoint.
Again, in the experiment of rolling a die, consider the events A 'an odd number appears' and event $B$ 'a number less than 4 appears'.
Obviously, $A=\{1,3,5\}$ and $B=\{1,2,3\}$
Now, $3 \in A$ as well as $3 \in B$
Therefore, $A$ and $B$ are not mutually exclusive events.
Remark Simple events of a sample space are always mutually exclusive.